Falling Dominoes:
A Theory of Rare Events and Crisis Contagion∗
H ENG C HEN
University of Hong Kong
W ING S UEN
University of Hong Kong
December 15, 2014
Abstract. Crises, such as revolutions and currency attacks, rarely occur;
but when they do they typically arrive in waves. The rarity of crises itself is an important contagion mechanism in a multiple-country dynamic
global game model. When players are uncertain about the true model of
the world, observing a rare success elsewhere can substantially change
their expectations concerning the payoffs from attacking or defending the
regime. Such dramatic revisions in beliefs, amplified by strategic complementarity in actions, may lead to a series of attacks in other countries. The
crisis period can be long-lasting, but will eventually come to an end.
Keywords. spread of revolutions, crisis contagion, model uncertainty, confidence collapse
JEL Classification. D74, D83, D84, G01
∗ Chen:
School of Economics and Finance, University of Hong Kong, Pokfulam, Hong Kong (email:
hengchen@hku.hk); Suen: School of Economics and Finance, University of Hong Kong, Pokfulam,
Hong Kong (email: wsuen@econ.hku.hk.). We thank Tiancheng Chen, Shangguan Ruo, and Fanqi Shi
for their excellent research assistance.
1.
Introduction
Crises are typically infrequent events but they tend to be contagious, sometimes even
among countries or regions that are seemingly unrelated. These two salient features
are particularly evident for currency attacks, mass revolutions, and bank runs. In
this paper we offer a theory to explain the link between the rarity of crises and the
contagion phenomenon.
Eichengreen, Rose, and Wyplosz (1996a) offer systematic empirical evidence which
corroborates these features for currency crises. With a panel of quarterly data covering
twenty industrial economies from 1959 to 1993, they can only identify a small sample
of 77 crises out of 2800 total observations.1 They estimate a binary probit model and
find robust evidence for contagion: controlling for a wide range of macroeconomic
variables, the existence of a currency attack elsewhere raises the probability of an attack on the domestic currency.2 Figure 1(a) shows the number of currency attacks in
each quarter during the operation of European Monetary System (EMS).3 Three distinct clusters of attacks are evident in this figure.
6
20
16
4
12
8
2
4
0
1979
1982
1985
1988
1991
0
1970
1994
(a) Currency crises in EMS
1978
1986
1994
2002
2010
(b) Mass revolutions
Figure 1. Rarity and Contagion.
Revolutions as a mass movement are also a relatively rare occurrence (Walt 1992),
and they typically arrive in waves (Katz 1997).4 An investigation of revolutions after
1970 reveals that most of them clustered in three waves: the collapse of communism,
1 They compile a speculative pressure index, which is a weighted average of percentage changes in
exchange rate, foreign reserve and interest rate. Currency attacks are identified as quarters in which the
index is at least one and a half standard deviations above the sample mean.
2 Other empirical evidence and anecdotes for contagion of speculative attacks abound. See, for example, Bordo and Murshid (2000).
3 EMS (Stage I) is chosen because it has well defined operating time and member states. Crisis data
are obtained from Eichengreen, Rose, and Wyplosz (1996b).
4 Following Kuran (1989), political revolution is defined as political regime change brought about in
a short period of time through a massive action against the status quo with popular participation.
1
the “color revolutions,” and the Arab Spring. See Figure 1(b).5 A similar pattern is
also evident for bank runs.6
Another interesting observation about contagion is that the spread of crisis is not
necessarily confined to countries with close economic linkages or political ties. For
instance, it is difficult to justify the transmission of crisis from Russia to Brazil or from
Mexico to Argentina with any real linkage between these countries (Bordo and Murshid 2000, Krugman 1999). Similarly, “revolutionary spirit” often ripples across countries that seem to be totally unconnected. For example, the chain of revolutions in Eastern Europe in 1990 spurred mass movements and led to political changes in Africa.7
Strikingly, the call for “Jasmine Revolution” that started in Tunisia and inspired Arab
countries even echoed in China, where it stirred protests in a handful of major cities,
including Beijing and Shanghai.8
In this paper, we develop a theory to explain why rare crises tend to cluster and
why crises can spread among countries that are seemingly unconnected either economically, financially, or politically. Toward this end, we construct a multiple-country
dynamic global game model, which is general enough to address the similar pattern
for speculative attacks, political revolutions, and bank runs.9
The key element of our theory is that people are uncertain about how the world
works, but they can learn it from observing equilibrium outcomes in other countries.
This feature of “model uncertainty” is captured by assuming that players maintain two
hypothetical descriptions of the world, but cannot be completely sure about which one
is true. One of the worlds is tranquil: winning a battle against autocratic regimes or
central banks is particularly difficult. The other world is frantic: overthrowing the
regime is relatively easy and successes are commonplace.
Suppose that the true world is indeed tranquil and players believe so with a probability close to one. In this situation, success takes place with a very low probability.
When a rare success is actually observed elsewhere, the little doubt harbored by these
5 We
perform the “runs test” to test the randomness or serial dependence of elements in this sequence. We find the null hypothesis that these observations are independent can be rejected at the five
percent significance level. To perform the test conservatively, we set all the positive values in the sequence to 1. If we start the period of investigation from 1980 or exclude the period of the Arab Spring,
we can still reject the null hypothesis. We thank Nandiyang Zhang for making this data set available to
us.
6 In the United States, the median time interval separating nationwide banking panics was 16 years
in the pre-Federal Reserve era, and even fewer panics existed in the post-Federal Reserve period (Jalil
2013). It was rather unusual to observe a standalone bank run that did not affect other banks. In fact,
the commonly used working definition of banking panics is a cluster of bank runs and suspensions.
7 See “Wind of change, but a different one,” The Economist, 14 July 1990.
8 See “Calls for a ’Jasmine Revolution’ in China persist,” The New York Times, 23 February 2011.
9 Morris and Shin (1998) develop the global game framework to address issues related to currency
crises. The framework has been adapted to model political regime changes (Edmond 2013; Chen, Lu,
and Suen 2014) and bank runs (Goldstein and Pauzner 2005).
2
Bayesian players that the true world might be frantic grows dramatically and their
confidence in the world being tranquil collapses. As a consequence, citizens in the second country will be more aggressive in attacking the regime—not only because they
feel that success is more likely, but also because they think their fellow citizens will
reason in the same way. Anticipating a massive attack, the regime believes it is less
worthwhile to defend itself. Another collapse in the second country would deepen
this crisis further: the regime in the third country collapses with an even higher probability. However, given the true world is tranquil in nature, the law of large numbers
eventually ends the “frantic period,” and public confidence recovers or revolutionary
euphoria fades as the number of failed attacks increases over time.
We should emphasize that our theory spotlights the role of rarity: only rare crises
can shake the public confidence dramatically. If successful attacks are common, this
learning mechanism does not produce any quantitatively sizable contagion effect. Furthermore, the contagion mechanism we describe here is effective even for countries
with uncorrelated fundamentals, as long as people believe that the same model of the
world applies. Citizens believe that a rare crisis in one country may signal systematic weaknesses in other countries that bear institutional similarities to the originating
one, because these countries are supposed to operate in a similar fashion with the same
rules of the game.10
While much has been written about the contagion of currency crisis, there is little
research on why revolutions often occur in waves. Political scientists have noticed the
need for a contagion model to explain the spread of revolution (Francisco 1993). Our
contagion model can be applied to both situations. Kuran (1997) argues that revolution
contagion is caused by the “domino effect,” as citizens in one country are inspired by
revolutions in other countries. However, a concrete link is yet to be specified between
actions taken in one country and those in others. Our paper characterizes exactly how
such inspiration can be modeled as Bayesian updating following rare events, and how
strategic complementarities reinforce the effects of belief revisions.
In this model, one successful revolution or speculative attack may lead to rampant
collapse of confidence or outbreak of euphoria and, as a result, a series of attacks in
other countries. Such a prediction is supported by surveys conducted among market
participants. Eichengreen and Wyplosz (1993) report that more than 90 percent of the
respondents agreed that the Exchange Rate Mechanism crisis of 1992–93 was contagious, and 77 percent of them attributed the contagion phenomenon to the reason that
“the markets had ’tasted blood’ [realized that there were profits to be made].” That
10 This
implication is consistent with empirical findings in studies on the contagion of currency crisis.
For example, Dasgupta, Leon-Gonzalez, and Shortland (2011) show that crises tend to spread among
economies which are institutionally similar.
3
is, after observing an encouraging success, market participants believed that it would
be very easy to conquer another currency and make profit. Similarly, revolutionary
euphoria is often stimulated after a success is observed in neighboring countries. For
example, the Tunisia revolution was often considered as one such catalyst which inspired people in other countries and ignited the Arab spring.11
Economists, commentators, and policy makers have long maintained that psychological factors may be critical in the international transmission of crises. Kindleberger
and Aliber (2011) argue that “there are psychological connections, as when investor
euphoria or pessimism in one country affects investors in others.” They regard specifically the spread of crisis from Russia to Brazil as largely psychological. Among many
others, Reinhart and Rogoff (2009) also blame the “fickle nature of confidence” for
contagion of currency crises. We adopt the position that the so-called “psychological” factors can be rationalized. In our model, we formalize the effect of “euphoria or
pessimism” spillover, with all the players being Bayesian learners.
In economics, multiple equilibria models with self-fulfilling feature, in the spirit of
Diamond and Dybvig (1983) or Obstfeld (1986), are typically considered convenient
to shed light on the contagion issue (Masson 1999): the success of currency attack in
a neighboring country triggers the domestic economy to “jump” to a bad equilibrium
where everybody frantically attacks. However, the missing part in this argument is
how equilibrium is selected. By adopting a global game framework, our model does
not exhibit multiple equilibria. Nevertheless the evolution of beliefs over different
models of the world provides an explicit link to determine the switch between periods
of tranquility and periods of frenzy.
This model differs from Drazen’s (2000) theory of political contagion, which holds
that the outbreak of crisis in a member country of a government “club” lowers the
value of its membership and therefore undermines other member governments’ commitment to the fixed rate arrangement. Our model features the decision of governments to defend the regime. Upon observing a crisis elsewhere, a government is more
likely to give up defending as it expects larger scale of attack, which in turn invites
even more attackers.
Our model also differs from contagion models that feature herding or information
cascades (e.g., Calvo and Mendoza 2000). We allow players to observe equilibrium
outcomes in precedent countries, but not actions of other individuals or aggregate
action. More importantly, our model does not rely on having players discard their private information to follow their predecessors’ actions. The private beliefs of all players
11 Anecdotal accounts and media coverages which document this phenomenon abound.
For instance,
see “Thousands Protest across Egypt, Inspired by Tunisia,” NPR, 25 January, 2011; and “Arab Revolutions: from Tunisia to Egypt, is this the Beginning of a Trend?” The Huffington Post, 1 February, 2011.
4
are incorporated into the equilibrium outcome, but when an unexpected outcome occurs the beliefs of subsequent sets of players are altered dramatically.
This paper is also different from the dynamic coordination games of Angeletos,
Hellwig, and Pavan (2007) and Chamley (1999), in which the same set of agents extract
information about a latent variable (i.e., the fundamental) from history and take action
repeatedly in each period. In our model, players in each country act only once, and
they are uncertain not only about a latent variable but also a latent structure (i.e., the
way the world operates). The observed history is only useful for updating the belief
about the latent structure. Therefore, the dynamics are completely captured by the
evolution of the belief over the model space. In one extension of this model, we also
show that our mechanism still works even when we allow for heterogeneity in the
latent structure across countries; see Section 4.2.
2.
The Model with One Country
In this section we lay out the benchmark model with a single country and expound
its qualitative properties. Section 3 characterizes the linkage between countries and
details how the rarity of crises causes contagion.
2.1.
Players and Payoffs
Consider an economy populated by a unit mass of ex ante identical agents, indexed
by i ∈ [0, 1], who play against another player, the regime. Agent i chooses one of
two actions: attack (ai = 1) or not attack (ai = 0). The aggregate mass of attackers is
denoted A. Simultaneously, the regime can also choose to defend (y = 1) or not defend
(y = 0).12 If the regime defends, the attack can be either a success (S = 1), where the
regime is forced to surrender or to devalue the domestic currency, or a failure (S = 0),
where the status quo is maintained. If the regime chooses to give up fighting, the
attack is always successful (S = 1), regardless of the size of the attack A.
An agent’s payoff depends both on whether the attack is successful and on whether
the agent chooses to attack. A positive cost, c ∈ (0, 1), has to be paid if she attacks.
Upon a success, agents who attack receive a benefit, b = 1, and those who do not
participate receive no benefit.13 If the regime defends, it has to pay a country-specific
12 As
in much of the literature, we assume that both players move at the same time. The alternative
scenarios where the regime takes action before or after agents attack, have been dealt with in Angeletos,
Hellwig, and Pavan (2006) and Goldstein, Ozdenoren, and Yuan (2011), respectively.
13 In the context of revolution, this payoff structure abstracts from free-riding issues, which is common
in the literature that models revolution as coordination games, e.g., Bernhardt and Shadmehr (2011) and
Edmond (2013). A more general payoff structure can incorporate the fact that the fruits of a revolution
are a public good, with a sufficient condition that will ensure that citizens still have incentives to act
against the regime despite the free-riding problem. Once such a condition is satisfied, the concern
of free-riding only increases the opportunity cost of participation but does not change the qualitative
5
cost d.14 The payoff to the regime is V if it survives the attack; and 0 otherwise. If the
regime does not defend, it collapses and the payoff to the regime is normalized to 0.
The net utility u for agent i and the net utility v for the regime are:
u ( ai , S ) =
2.2.



1−c


−c



0
if ai = 1 and S = 1,
v(y, S) =
if ai = 1 and S = 0,
if ai = 0;



−d


V−d



0
if y = 1 and S = 1,
if y = 1 and S = 0,
if y = 0.
Information Structure
Let θ represent the strength of the regime. Citizens are ex ante identical and become
ex post heterogeneous after each of them observes a noisy private signal about the
strength of the regime,
xi = θ + ε i ,
where the strength θ is drawn by nature from a normal distribution, N (µ, σ2 ), and
the idiosyncratic noise ε i ∼ N (0, σx2 ) is independent of θ, and is independently and
identically distributed across i. In contrast, the regime can observe perfectly the fundamental θ, which is selected by nature.15 Moreover, the regime knows the cost of
defending d, while agents cannot observe it but know that d is drawn from a distribution function H on the support [0, d] where d ≤ V.
The main departure from a standard global game is that, besides the uncertainty
about regime strength θ, the rule of the game is also not common knowledge. In other
words, we allow for model uncertainty—neither the regime nor the agents are certain
about how the world operates. Specifically, we assume that players maintain two
alternative hypotheses: the world can be either a tranquility world, denoted T, or a
frenzy world, denoted F. They also know that one of the two is the true model for
how the world operates. In Section 4.3, we show that our results continue to hold in
the general case where there are a wide range of hypothetical descriptions about the
world, with one of them being the true description.
Tranquility and frenzy worlds differ only in the success determination condition
features of the model.
14 To keep a currency peg typically requires sacrificing domestic goals, which are in conflict with a
fixed exchange rate. Such a cost can vary across countries. In a context of revolution, costs have to
be incurred to maintain a repressive regime if the regime chooses to defend. Such costs include, but
are not confined to, the cost of maintaining troops, as repressive policies may have negative implications for economic performance. In Section 4.3, we discuss a case where the defense cost are the same
across country but unknown to agents. We show that the contagion mechanism that we propose will
be strengthened.
15 This assumption has been widely adopted in the literature, e.g., Hellwig, Mukherji, and Tsyvinski
(2006). Our results do not rely on this specific informational asymmetry between the regime and agents.
See discussion on this in Section 4.3.
6
when the regime chooses to defend and confront the attackers. Specifically, it is easier
to overcome the regime’s defense in F-world than in T-world: the outcome of the
confrontation is determined by
S j ( A, θ ) = 1 A(θ ) ≥ D j (θ ) ,
where D j (θ ) is an increasing function of θ, and DT (θ ) ≥ DF (θ ) for any θ.
One may interpret the possibility of these two worlds as uncertainty about the
defense technology. In the context of mass revolution, θ may stand for the size of
troops defending it. Even holding it constant, there is some uncertainty about how big
a revolt is needed to topple the regime: in a frenzy world a small mass of attackers is
sufficient to produce a revolt that would overcome the troops, while in a tranquility
world a much larger mass of attackers is required. An alternative interpretation is that
there is uncertainty about the consequences of attack. In the context of currency crisis,
θ may stand for the amount of foreign reserves. An attack on currency may produce
unpredictable economic pains on the real sector in addition to the depletion of foreign
reserves.16 In a frenzy world these economic pains may be so large that the central
bank devalues the currency well before its reserves are exhausted; in a tranquility
world the central bank can hold on to the fixed exchange rate for much longer.
Both the regime and the agents share a common prior belief π that they live in
tranquility world. This section simply treats π as exogenous and studies its role in
equilibrium via comparative statics. In the next section we characterize the evolution
of this belief, which is the key to the contagion mechanism in our model.
2.3.
Equilibrium
Definition 1. An equilibrium consists of a set of posterior beliefs which are derived using
Bayes’ rule, an attack decision a( xi , π ), a defense decision y(θ, π, d), and a mass of attackers
A(θ, π ), such that
y(θ, π, d) = argmax {[π (1 − ST ( A, θ )) + (1 − π )(1 − S F ( A, θ ))]V − d} y;
y∈{0,1}
a( xi , π ) = argmax
ai ∈{0,1}
Z ∞
Z
∞
−∞
πu( ai , ST ( A, θ )) + (1 − π )u( ai , S F ( A, θ ))
1
A(θ, π ) =
a ( xi , π ) φ
σx
−∞
xi − θ
σx
f (θ | xi ) dθ ;
dxi ,
where f (θ | xi ) is the posterior density of θ.
16 For example, the increased cost of servicing the public debt is one reason why a government accedes
to devaluation pressures. Obstfeld (1994) offers a description of different types of economic pains that
the real sector in a country under attack may suffer.
7
We focus on monotone equilibrium throughout (which is shown to be the only
equilibrium). In such an equilibrium, there exists a cutoff x ∗ such that agent i attacks
the regime if and only if xi ≤ x ∗ . An agent who observes a low value of xi believes
that the strength of the regime is low, which raises the expected payoff from attacking
the regime. In a monotone equilibrium, the size of the attack A(θ, π ) decreases in θ.
When the regime chooses to defend, there exist thresholds θ T∗ and θ F∗ such that the
attack would be successful if and only if θ ≤ θ j∗ in world j.
Given our assumption about the attack technology, the regime is more likely to
be toppled when confronting attackers in a frenzy world than in a tranquility world.
Thus, it always holds that θ T∗ ≤ θ F∗ . When θ is above θ F∗ , the attack would fail as
long as the regime chooses to defend. Since V > d, the regime always chooses to
defend in this case. When θ is below θ T∗ , defending the regime is futile because the
attackers would succeed despite the effort of the regime to maintain the status quo. So
the regime always chooses not to defend in this case. When θ is between θ T∗ and θ F∗ ,
defense is effective only if it is a tranquility world. Consequently, the regime adopts
the following monotonic decision rule:
y(θ, π, d) =



1





0
if θ ≥ θ F∗ ,
or θ ∈ [θ T∗ , θ F∗ ) and πV ≥ d,
otherwise.
Let Pj be the probability of success from the perspective of an agent with private
information x if she believes that the true world is j. Then,
!
PT (θ T∗ , θ F∗ , x, π ) = Φ
θ T∗ − X
p
βσ
!
PF (θ F∗ , x, π ) = Φ
θ F∗ − X
p
βσ
+
Z θ∗
F
θ T∗
1
(1 − H (πV )) p φ
βσ
t−X
p
βσ
!
dt,
;
where X = βx + (1 − β) µ is the posterior mean of θ, β = σ2 /(σx2 + σ2 ) is the weight
that the agent attaches to her private information, and Φ is the standard normal distribution function. The first term in PT represents the probability that θ is less than
θ T∗ (in which case the regime never defends and the attack is always successful). The
second term is the probability that the regime gives up (the attack would have failed
if the regime chose to defend). The probability that the regime collapses in the frenzy
world, PF , is simply given by the probability that θ is less than θ F∗ (because in this case
the attack is successful regardless of whether the regime defends or not).
A marginal agent with private information x ∗ is indifferent between attacking or
8
not attacking. For such an agent, the probability of success is a weighted average of PT
and PF and the weight attached to PT is π. In other words, the indifference condition
can be written as:
c = πPT (θ T∗ , θ F∗ , x ∗ , π ) + (1 − π ) PF (θ F∗ , x ∗ , π ).
(1)
The equilibrium regime survival thresholds must satisfy the following critical mass
conditions:
∗
x − θ T∗
Φ
= DT (θ T∗ ),
(2)
σ
∗ x ∗
x − θF
= DF (θ F∗ ).
(3)
Φ
σx
A monotone equilibrium can be characterized by the triple ( x ∗ , θ T∗ , θ F∗ ) that solves the
system of equations (1), (2) and (3).
It will also be useful to obtain the “objective” probability (i.e., the probability for
one with no private information about θ or d) that the regime collapses. Let p T and p F
represent this probability in the tranquility world and in the frenzy world, respectively.
We have
∗
Z θ∗
F
θT − µ
1
t−µ
pT = Φ
+
dt;
(4)
(1 − H (πV )) φ
σ
σ
σ
θ T∗
∗
θF − µ
pF = Φ
.
(5)
σ
Proposition 1. If D 0j (θ )φ(0)σx < σ2 for j = T, F, a monotone equilibrium exists and is the
only equilibrium. Further, x ∗ , θ T∗ , and θ F∗ , as well as p T and p F , decrease in π.
Proof. See the Appendix.
From equations (2) and (3), it is easy to see that we must have θ T∗ ≤ θ F∗ in a monotone equilibrium, because DF (θ ) ≤ DT (θ ) for all θ and both functions are increasing.
Since θ T∗ ≤ θ F∗ , equations (4) and (5) also imply that we must have p T ≤ p F . Likewise,
for any agent with private information xi , we have PT (θ T∗ , θ F∗ , xi , π ) ≤ PF (θ F∗ , xi , π ).
In other words, the probability of a successful attack, either subjective or objective, is
lower in a tranquility world than in a frenzy world.
A decline in public belief π has two effects. First, since PT ≤ PF , equation (1) implies that it raises the expected payoff from attacking. Second, when θ is between θ T∗
and θ F∗ , the expected payoff for the regime from fighting against attack, πV, also falls.
As agents expect the regime to give up defending more often, this raises PT and, by
9
equation (1), raises the expected payoff from attacking. Consequentially, both effects
cause agents to become more aggressive, i.e., x ∗ increases. These effects are strengthened through complementarity in action between agents and the regime: expecting an
attack of larger scale, the regime gives up with higher probability; anticipating this,
agents attack even more aggressively. Moreover, complementarity in action among
the agents themselves as in a standard global game model further amplifies these effects: an increase in x ∗ raises the critical success thresholds θ T∗ and θ F∗ , which in turn
increases the payoff from attacking and causes x ∗ to rise further. Therefore, the attack
is more likely to be successful, no matter the true world is tranquility or frenzy. That
is, both p T and p F increase.
3.
3.1.
Contagion across Countries
The Evolution of Public Belief
Now we turn to a dynamic multi-country version of the benchmark model. Countries are identical except that the fundamental strength of the regime θ and the cost of
defending d for each country are independent draws.17 They are lined up in a fixed
order and take action sequentially.18 We use subscript t on some variables to indicate
the country concerned.
Players in a country can observe whether attacks in preceding countries are successful or not and update their beliefs accordingly. We let πt represent the public
belief in country t that the world is a tranquility world. This is the belief of a player
in country t after observing the history of successes in preceding countries but before
observing her private information (xi for agents or θ for the regime).19 By Bayes’ rule,
the belief updating process can be written recursively as:


π
p T (πt )
t
π t +1
1− π t p F ( π t )
=
 π t 1− p T ( π t )
1 − π t +1
1− π t 1− p F ( π t )
if St = 1,
if St = 0.
(6)
In equation (6), p T /p F is the likelihood ratio of observing a successful attack in country t. It should be noted that the equilibrium values of p T and p F depend on θ T∗ and
17 We
share the view of Drazen (2000) that contagion is “the phenomenon whereby a currency crisis
itself in one country makes a currency crisis in another country more likely.” It is conceptually distinct
from common shocks that affect all countries and is also different from the so-called “spillover effect”
in which the fundamental in one country affects those in others.
18 This assumption is intended to capture the usual pattern of how crises diffuse: they do not occur
at different countries simultaneously. It bears similarity to standard herding models (Banerjee 1992 and
Bikhchandani, Hirshleifer, and Welch 1992), where the timing of actions is fixed. To focus on the key
mechanism of contagion, we abstract from the endogenous timing of actions.
19 Observations on the outcomes of previous countries do not help agents to update their belief on the
fundamental in their own countries, because the fundamentals are uncorrelated across countries.
10
θ F∗ , which in turn depend on πt . Likewise, the likelihood ratio of observing an unsuccessful attack in country t is (1 − p T )/(1 − p F ), which is also a function of πt . The
evolution of public belief described by equation (6) depends on the stochastic process
{St }. Assume that the true world is a T-world. The probability law governing this
stochastic process is:
St =

1
with probability p T (πt ),
0
with probability 1 − p T (πt ).
(7)
Recall that an attack is more likely to be successful in a frenzy world than in a tranquility world, i.e., p T /p F < 1. Therefore, a successful attack in country t is evidence
in favor of the hypothesis that the world is frenzy. As a result, the public belief πt+1
becomes lower than πt . But a lower πt+1 in turn causes agents to become more aggressive and the regime to defend less often in country t + 1. It follows from Proposition
1 that the objective probability p T of a successful attack in country t + 1 increases following a successful attack in country t. This link provides a mechanism for crises to
be contagious across countries.
3.2.
Rare Event and Collapse in Confidence
Since a successful attack always causes players to lower their belief that the world is
tranquil, the contagion mechanism described in the previous subsection always tends
to produce positive serial correlation in the occurrence of crises. However, the quantitative significance of this serial correlation critically depends on the magnitude of belief
updating. If the likelihood of success is similar in the two worlds (p T is nearly as large
as p F ), then even when a success is observed the amount of downward revision in
belief is tiny. In contrast, if p T is small, crises will be rare in a tranquility world. When
the likelihood ratio p T /p F is small, the occurrence of a crisis will trigger a substantial downward revision in belief. Suppose that the difference in the defense technology D j (θ; k j ) is characterized by the parameter k j , where j = T, F. To ensure that the
rareness of crises can be obtained in T-world, we let DT goes to infinity when k T goes
to infinity. One example is a linear defense technology with D j (θ ) = kθ + k j , where k
is positive and k T > k F .20
Proposition 2. As k T increases, the regime survival threshold θ T∗ in world T falls without
bound, while θ F∗ and x ∗ decrease but remain bounded. For k T sufficiently large, an increase in
k T causes (a) success to become rarer in world T (i.e., p T decreases); and (b) a greater downward
revision in the belief in world T upon observing success (i.e., p T /p F decreases). In the limit,
20 Any alternative assumptions that ensure the rarity of crises can also work and would not change
our results qualitatively.
11
the belief in country t + 1 upon observing St = 1 is
lim πt+1 =
k T →∞
πt (1 − H (πt V ))
.
πt (1 − H (πt V )) + 1 − πt
Because the likelihood ratio p T /p F approaches 1 − H (πV ) as k T goes to infinity,
the right tail of the distribution of d is relevant for belief updating. We say that H has
a thinner right tail than the uniform distribution on [0, V ] if there exists dˆ such that, for
all d > d,ˆ
d
1 − H (d) < 1 − .
V
Ex ante, a regime is less likely to incur high defense cost when H has a thin right tail.
This makes it more likely to successfully defend itself against attack in the tranquility
world, which helps make both p T and p T /p F small. The next result follows immediately from the formula for πt+1 stated in Proposition 2.
Corollary 1. For any e > 0, there exists k T such that for all k T > k T , the updated belief πt+1
following St = 1 satisfies:
1. If H has support on [0, d] with d < V, then for any πt ∈ (d/V, 1), πt+1 < e.
2. If H has a thinner right tail than the uniform distribution on [0, V ], then for any πt ∈
ˆ
(d/V,
1), πt+1 < πt /(1 + πt ) + e.
Case (1) of Corollary 1 says that the public belief can drop from near 1 to near 0
following a single successful attack, provided d < V. Even when this condition is not
satisfied, a rare success can trigger a substantial collapse in belief under the relatively
mild assumption that the distribution H has a thin right tail. For example, in case (2)
ˆ
of Corollary 1, if πt = 0.9 > d/V,
then πt+1 < 0.48 following a successful attack in
country t.
When k T is very large, success is expected to be difficult to achieve in the tranquility
world. Once players attach a high probability to the tranquility hypothesis, few agents
would attack and the regime would defend more often. As a result, success indeed
occurs with a very low probability. The larger is k T , the rarer is success. But when
such a rare success happens, agents drastically lower their belief that they live in the
tranquility world, precisely because they do not expect such an event to occur if their
maintained hypothesis is true.
The consequence of a rare success elsewhere and the associated dramatic belief
revision is described in Proposition 1. Agents become much more aggressive and
decrease their estimates of regime’s survival substantially, and the regime is also less
12
confident in defense. As a result, the probability of the regime collapsing increases
sharply after the first domino falls. In other words, a tranquil world may suddenly
become frantic just because people’s beliefs change.
The transmission mechanism we describe here captures popular notions about crisis contagion often heard in policy discussions: a crisis elsewhere leads to “abrupt
shifts in investor confidence” (Pesenti and Tille 2000) and weakens the domestic government’s “commitment to a fixed exchange rate” (Krugman 1999). Economists tend
to view such abrupt changes through the lens of multiple equilibria, but remain silent
on how agents shift from one equilibrium to another. In this unique equilibrium
model, as long as people are not completely sure about their model of the world,
Bayesian belief updating can cause a sudden switch in behavior.
More importantly, our result emphasizes that the fragility of beliefs is closely related to the rarity of crises: it is precisely the unlikely occurrence of a crisis in a tranquility world that prompts people to fundamentally reassess whether the world is
indeed tranquil. In a model with complementarity between beliefs and actions, such
reassessment can become self-fulfilling, at least in the short run.
3.3.
Propagation and End of Crisis Period
It is important to point out that one successful attack in a country not only affects the
next country, but also triggers the plummeting of confidence for a number of subsequent countries and produces a sustained period of crisis.
Interestingly, although the period of turbulent time can be long-lasting, our model
predicts that in the long run it must come to an end. Intuitively, since the true world
is T, Bayesian learning implies that the public belief will on average recover after a
collapse in confidence. To see this clearly, write the belief updating equation (6) in
reciprocal form and take expectation conditional on the true world being T, we obtain:
1 − πt p F (πt )
1 − πt 1 − p F (πt )
1 − πt+1 T = p T (πt )
+ (1 − p T (πt ))
E
π t +1 πt p T (πt )
πt 1 − p T (πt )
1 − πt
=
.
πt
Thus, E[1/πt+1 | T ] = 1/πt . But since 1/π is strictly convex in π, Jensen’s inequality
implies
E [ π t +1 | T ] > π t .
Therefore, the public belief will eventually recover to a level close to one (until another
accidental success strikes) and the world becomes tranquil again. This feature of our
model is absent in most other models of contagion, which focus on how a crisis starts
13
and propagates, but pay less attention to how it ends.21
3.4.
Numerical Examples
The evolution of beliefs and equilibrium outcomes described by equations (6) and (7)
is not deterministic. In this subsection we provide numerical simulations to illustrate
how the occurrence of one rare event interacts with the contagion mechanism to produce a period of crises across countries.
For the benchmark exercise, we set the cost of attacking at c = 0.5. The variances
of private information and prior are the same, σx2 = σ2 = 1. The payoff for the regime
is V = 1 if it survives. The distribution H of the cost of defending is uniform on [0, d]
and d = V. We characterize the defense technology in the two alternative worlds
with a linear function, i.e., D j (θ ) = kθ + k j , and let k T = 5 and k F = −2, while
k = 3. The mean of the distribution of the fundamental is set to be µ = 0.5 in the
benchmark case. The parameters for T-world are chosen such that a successful attack
is a rare event: the regime collapses with a probability of 1.5 percent if the true world
is tranquil and if players fully believe so. The parameters for the F-world would imply
that the regime would collapse with high probability if players fully believe in the Fworld. We assume that players are very confident that the true world is tranquil, but
they are not 100 percent certain. We therefore set the initial public belief at π0 = 0.99.
In other words, they entertain a small possibility that the world is frantic.
We compute the dynamics of π (the public belief) and p T (the probability of successful attack in the true world) for thirty countries under two scenarios: there is a
success in the initial country (S0 = 1), or a failure in the initial country (S0 = 0). To
contrast the case where success is rare with the case where success is commonplace,
we repeat the exercise by setting µ = −1.5, so that success is common in the tranquility world as well (the probability of success would be 50 percent if players fully
believe that it is a T-world). We simulate each of the four sub-cases 2,000 times. After
averaging the outcomes of each country in these 2,000 histories, we obtain the average
sample path for π and p T .
Crisis, propagation, and slow recovery. The left column of Figure 2 shows the average
sample paths of π and p T in the case where crises are rare. An accidental success
in country 0 shakes the public confidence and unfolds a “turbulent time”: π drops
from 99 percent to 78 percent, and the probability of success in country 1 shoots up
from less than 1.5 percent to more than 13.5 percent. If we start the simulation with
21 In
a currency attack model with multiple equilibria, for example, one may attribute the beginning
of a crisis to individuals suddenly jumping to a bad equilibrium altogether (Masson 1999). But then one
must argue that they always coordinate to jump back to the good equilibrium to end the crisis.
14
π
1.0
π
1.0
0.9
0.9
0.8
0.8
t
0.7
0
5
10
15
20
25
t
0.7
30
0
(a) The dynamics of πt (µ = 0.5)
5
10
15
20
25
30
(b) The dynamics of πt (µ = −1.5)
pT
pT
0.20
0.65
0.15
0.60
0.10
0.55
0.05
t
0
0.50
t
0.45
0
5
10
15
20
25
30
0
(c) The dynamics of p T (µ = 0.5)
5
10
15
20
25
30
(d) The dynamics of p T (µ = −1.5)
Figure 2. Contrasting Patterns of Contagion when Success is Rare or Commonplace.
The solid lines stand for the scenario with S0 = 1, and the dashed lines for S0 = 0. The two graphs in
the left column display results from the case where success is rare, with µ = 0.5, while the right column
shows the case where success is commonplace, with µ = −1.5.
lower values of π0 , or with a distribution H that has a thinner tail than the uniform
distribution, then our quantitative results will be even more dramatic.22
The sustained crisis period following a rare success is evident in Figures 2(a) and
2(c), where we see that π and p T do not revert to their baseline levels even after thirty
countries take actions. One reason of the observed slow recovery is that the belief
updating process is not symmetric. In our example, the public belief changes from
π0 = 0.99 to π1 = 0.78 after a successful attack in country 0. But even if the attack
in country 1 fails, the public belief only goes back up to π2 = 0.87, still considerably
lower than the initial level of 0.99.
This asymmetry is caused by the rarity of success. In country 0, a successful attack
is fifty times more likely in an F-world than in a T-world (p T /p F = 0.027). Success
in the initial country is very strong evidence against tranquility, and hence triggers a
large downward revision is belief. In country 1, a failed attack is two times more likely
in a T-world than in an F-world ((1 − p T )/(1 − p F ) = 1.99). This means that failure
in the subsequent country is only mild evidence against frenzy, and hence it triggers a
mild rebound in confidence.
22 For
example, if π0 = 0.95 and S0 = 1, then π1 = 0.59 and p T for country 1 is more than 26 percent.
15
Another effect which contributes to the slow recovery is that, with a significantly
higher p T in country 1, the probability becomes larger that the public belief will go
down even further following another success in country 1. Thus, when we focus on
the average sample path, the public belief in country 2 is on average π2 = 0.82. See
Figure 2(a).
The dashed line in Figure 2(a) presents the case where the rare success does not
occur in country 0. In this case the average sample path for πt monotonically increases,
since failure is evidence for tranquility. But the increase is almost not discernible,
because failure is so common that the additional evidence does not have much effect.
Commonplace success. The right column of Figure 2 shows the average sample paths
of π and p T for the case where success is commonplace. In this case, one successful
attack in the initial country still triggers a fall in confidence and an increase in the
probability of success in subsequent countries, but the effects are very small. When
successes are commonplace even in the true world, the occurrence of a successful attack in the initial country is only weak evidence in favor of the frenzy world hypothesis. In our example, the public belief only goes down from π0 = 0.99 to π1 = 0.98.
As a result, the effects on the behavior of players in other countries are negligible. The
contrast between this case and the case of rare events shows that it is the combination
of rarity and the contagion mechanism that produce the clustering of crises.
Crisis deepening. As our theory predicts, a rare successful attack in country 0 increases the chance of another success in country 1 substantially. If such a success happens to materialize in country 1 again, people’s confidence in a tranquility world falls
further and the probability of observing subsequent successes will be even higher.
Figure 3 shows how the crisis can deepen after more than one success in a row. The
dashed line shows the situation for the case with S0 = S1 = 1. Following another
success in country 1, the public belief is further lowered to π2 = 0.46 and p T shoots up
to 35 percent.
The dotted line shows the case with S0 = S1 = S2 = 1. Observed that after two
successes in a row, the third success leads to a much smaller revision in public belief.
That is because the probability of success has gone up to 35 percent, so that successes
are no longer rare. Thus, further successes can only produce a small effect on beliefs
and hence on actions.
Runs tests. One of the main predictions of our model is that successful attacks tend
to occur in serial clusters: a success in one country raises the probability of success
in the next country. A commonly used method to detect such serial dependence of binary events is the runs test (Wald and Wolfowitz 1940). For each of the 2,000 simulated
16
π
1.0
pT
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
t
0
0
5
10
15
20
25
30
t
0
0
5
10
15
20
25
30
(b) The dynamics of p T
(a) The dynamics of πt
Figure 3. Crisis Deepening.
The average sample paths of π and p T after one, two or three consecutive successes. Solid line is the case
for S0 = 1, dashed line for S0 = S1 = 1, and dotted line for S0 = S1 = S2 = 1.
histories of successes and failures, we group all consecutive successes or consecutive
failures into a “run” and count the total number of runs in each history of thirty countries. When successes tend to be clustered, the number of runs will be relatively small
compared to the expected number.23 Thus, if the runs test statistic is significantly negative, it is evidence in favor of the hypothesis that successes are positively serially
correlated. For each of the four scenarios, we compute the proportion of simulated
histories that fails the one-tailed runs test at the 5 percent significance level. Table 1
summarizes the results.
Table 1. The Runs Test.
Successes are rare (µ = 0.5)
Successes are commonplace (µ = −1.5)
S0 = 1
S0 = 0
61.2%
4.6%
3.9%
5.5%
Note: Histories with all failures are excluded because they pass the
test automatically (the runs test has no power).
In the rarity case, a predominant proportion (61 percent) of simulations fail the runs
test after observing S0 = 1. This indicates that an initial success in one country tends
to be followed by clusters of successes in other countries. In contrast, when S0 = 0,
only about 4 percent of the simulations fail the test, which is what one would expect
from pure sampling variability if there is no serial correlation.
In the case where successes are commonplace, the proportion of simulations that
fail the runs test after observing S0 = 1 is around 5 percent. This means that, although
the contagion mechanism is still present in this case, its effects are so weak that they
23 If there are
n successes and m failures in the sample, the expected number of runs is 2nm/(n + m) +
1 under the null hypothesis that successes and failures are drawn independently.
17
are not statistically discernible in small samples. Note also that observing success or
failure in the initial country does not matter much for the clustering of crises. In either
case, a statistician would be hard-pressed to detect the presence of contagion.
4.
4.1.
Discussion
Model Uncertainty and Correlation in Fundamental
Our contagion mechanism is purely informational. While acknowledging their importance, we abstract from those commonly assumed correlation of economic fundamentals and trade or financial market linkages which can cause and exacerbate contagion
across countries (e.g., Gerlach and Smets 1995; Allen and Gale 2000). Instead, the only
connection between countries is that they are subject to the same set of rules that govern the operation of the world. We show that even such a seemingly tenuous linkage
can be potentially a source of contagion, which is largely overlooked in the existing
literature.
How is our model distinguishable from an alternative model with correlation in
fundamentals across countries but without model uncertainty? Consider, for example,
a model in which the fundamental θt follows an autoregressive process of order 1, and
citizens in country t can observe a private signal about θt as well as the realized value
of θt−1 . Call this a correlation model. Obviously, citizens in country t are more likely
to attack (and the regime is less likely to defend) when θt−1 is low. Since θt is serially
correlated across countries, successes will also be serially correlated in the correlation
model. Moreover, provided that the long run average of θt is high, successes are rare
ex ante.
There are, however, subtle differences between our model and the correlation model.
First, our contagion mechanism relies solely on an informational linkage. If citizens
in country t do not observe the outcome of success or failure in country t − 1, then
equilibrium outcomes in different countries will be totally unrelated. In the correlation model, by contrast, outcomes will be correlated even if citizens in one country
do not observe what happened in another country. The correlation model imposes a
mechanical link that produces contagion, which does not depend on (but can be magnified by) the endogenous response of players. Second, as we stress, the contagion
mechanism would be not identifiable quantitively if crises are commonplace in our
model. In the correlation model, successes and failures are always dependent across
countries, no matter whether successes are rare or commonplace.24 Third, our model
24 We
specify, solve and simulate a model with correlation in fundamental. We perform the runs test
on the simulated data and conclude that there are always serial dependence in the sequence of outcomes
regardless of the ex ante probability of success.
18
Mass of attackers
Mass of attackers
0.05
0.05
t
0
0
5
10
15
t
0
20
0
(a) The correlation model
5
10
15
20
(b) The benchmark model
Figure 4. The Size of Attack in the Correlation Model and Benchmark Model.
The figure shows the average sample path of the size of attack based on 2,000 simulated histories. The
correlation model specifies an AR1 process with ρ = 0.5 and µ = 2. The fundamental strength in the
initial country is set to 0 to set off a successful attack. The parameters used in the benchmark model are
the same as those in Figure 2(a)
.
exhibits path dependence while the correlation model only exhibits state dependence, to
use the distinction made by Page (2006). In the correlation model, the outcome in
country t depends on other countries only through the state transition to θt ; the actual realization of successes or failures in other countries is irrelevant. In our model,
the entire history {S1 , . . . , St−1 } matters. The history {1, 0, 1}, for example, produces a
very different outcome for country 4 than does the history {0, 1, 1}.
In practice, when the underlying state is unobserved by the econometrician, it is
difficult to distinguish between path dependence and state dependence, especially
when the time series comprise only binary data about success or failure. The difference
becomes more apparent if the econometrician also has access to data on the size of
the attack in different countries. In the correlation model, the mass of attackers At
shoots up when the realization of the fundamental θt−1 is exceedingly low. Because
θt tends to regress back to the mean, the size of attackers declines on average in the
subsequent countries—regardless of whether the attack succeeds or fails in country
t. In our model, the mass of attackers At also shoots up once a success is observed
in the previous country, but what happens in subsequent countries depend on what
happens in country t—At+1 will be higher than At if St = 1 and will be lower than
At if St = 0. More interestingly, the size of attack on average can go up even further
in the subsequent countries, even though we show in Section 3.3 that the public belief
must decline on average.25 Figure 4 shows that our model and the correlation model
have very different implications for the mass of attackers following a success in one
country.
25 This
is possible because At depends on the public belief πt non-linearly.
19
We do not claim that the mechanism we expound here is the only one that matters, but our theory can shed light on situations that are otherwise difficult to explain
using direct linkages. Regarding currency attacks, the often-cited crisis transmission
from Russia to Brazil falls into that category. In the context of revolution, citizens can
learn about how a dictatorship state works from the crisis experience in other countries, even though the regime strength in each affected country may not necessarily be
correlated. Because revolutions are seldom observed, neither citizens nor the ruling
elites have a good understanding of the technology of revolutions, which justifies our
assumption on model uncertainty. Our model offers an interpretation of the revolution wave of the Arab Spring: the unexpected success in Tunisia triggered a large shift
in the belief on the revolution technology, which caused citizens to become more aggressive and some of the ruling elites to lose their will to suppress civilian uprisings,
and therefore ushered in a sequence of revolutions.
One natural implication of this model is that the degree of “similarity” between
countries (perceived by attackers) predicts the strength of our contagion mechanism,
provided that crises are rare. Countries would not affect each other if they were governed by rules that are entirely different. Conversely, if the rules of the game are
similar across countries, players can learn the robustness of these rules based on their
observations about successful or failed attacks against other countries. This prediction
is consistent with a number of empirical findings. For example, Eichengreen, Rose,
and Wyplosz (1996a) show that a crisis within the EMS had a larger contagion effect
on member countries than on non-member countries. Dasgupta, Leon-Gonzalez, and
Shortland (2011) find that crises tend to spread among countries which bear institutional similarities.
4.2.
Model Uncertainty and State Switching
In the benchmark model, we assume that the true world is indeed tranquil and agents
harbor only a slight doubt about it. Implicitly, we assume that countries with the same
institutional features, i.e., countries with fixed exchange rate system or countries with
dictatorship, may work in a similar fashion, even though the strength of each country
may not be correlated. In this extension, we assume that the state of the world can
actually shift from tranquility to frenzy and the other way around. Agents still only
observe the past outcomes but not the state of the world. In other words, we maintain
the assumption of model uncertainty and entertain the idea that countries may not
share the same set of characteristics.
Specifically, we adopt a Markov state-switching model. If the world in current
period is T, the chance to stay in T-world next period is q T ; if the world is F, the
20
Table 2. State Transition Matrix.
T in Period t + 1
F in Period t + 1
qT
1 − qF
1 − qT
qF
T in Period t
F in Period t
probability of staying in F-world next period is q F . See Table 2.
The belief about which world agents live in is summarized by the public belief πt .
Given the outcome in period t, the public belief is updated in the following fashion:
π t +1 =


πt p T
(1− π t ) p F
q + π p +(1−
(1 − q F )
πt p T +(1−πt ) p F T
πt ) p F
t T
π t (1− p T )
(1−π )(1− p )

q + π (1− p )+(t 1−π )(F 1− p ) (1 − q F )
πt (1− p T )+(1−πt )(1− p F ) T
t
t
T
F
if St = 1,
if St = 0.
Considering the following three cases. First, suppose both q T and q F are also close
to 1. This means that T-world rarely switches to F-world or vice versa. In this case, our
mechanism of model uncertainty is still at work. Suppose the public belief in country
t that the true state of the world is T is close enough to 1, a success would revise the
public belief downwards substantially. However, ex ante, the public belief gradually
converges to the long-run stationary distribution of the two worlds, which is
π∞ =
1 − qF
.
1 − qF + 1 − qT
When both q T and q F are close to 1, π∞ is close to 1/2, and the chance of success in
each world, p T (π∞ ) and p F (π∞ ) will be both substantially higher than 0. In other
words, in this model, crisis events are not rare in the long run.26 To illustrate, we
repeat the numerical exercise shown in Figure 2, by starting with T-world at t = 1 and
allowing the state of the world to transit from one to another, following the transition
matrix specified in Table 2. Figure 5 shows the results when q T = q F = 0.99. Notice
that our contagion mechanism is still at work even in this case when the initial belief
π0 is sufficiently close to 1, as evidenced by the large initial drop in public belief and
the subsequent recovery. However, also note that there is a long-run tendency for the
public belief to converge to its stationary state level.
Consider next the case where q T is close to 1 but q F is close to 0. In this case, the
long-run stationary probability of T-world is high. However, even if citizens in country t + 1 believe that the world has switched to a frenzy world in country t (following a
successful attack in that country, for example), they believe that their own country will
26 Likewise,
if q T is small but q F is big, then successes are not rare in the long run.
21
p
π
1.0
0.3
0.9
0.2
0.8
0.1
t
0.7
0
5
10
15
20
25
t
0
30
0
(a) The dynamics of π (q T = q F = 0.99)
5
10
15
20
25
30
(b) The dynamics of p (q T = q F = 0.99)
Figure 5. State Switching with Model Uncertainty.
Panel (a) repeats the numerical example of Figure 2(a), with the additional mechanism of Markov stateswitching. Panel (c) is the counterpart to Figure 2(c). This figure shows the case where both q T and q F
are close to 1.
have switched back to a tranquility world with high probability. Therefore, a success
observed in country t revises the public belief of agents in country t + 1 only a little.
In this case, crises are likely to be stand-alone incidents.
In the two cases mentioned above, either the rarity of success is not ensured in
the long run or contagion does not take place. Neither case is a good candidate for
explaining why rare crises tend to be contagious. Both features of rarity and contagion
can be present in a third case when q T is close to 1 and q F is not too big or too small
(e.g., q F = 0.7). In this case, the model is quantitatively similar to our benchmark
model, where a success in the pervious country revises the public belief downwards
dramatically, which gradually recovers and converges to a value below but close to
1. Therefore, we may consider our model as an approximation to this case where the
operation of the world can change. Even in this case, note that the key to contagion
is still the fact that agents cannot observe the true world, which is exactly the same
mechanism as we describe in the benchmark model. If instead citizens in each country
can observe the true world in the Markov state-switching model, then the probability
of success will simply fluctuate between a low level (when the world is tranquil) and
a high level (when the world is frantic). It will not exhibit the pattern of sudden fall
and gradual recovery shown in Figure 2(c) of our benchmark model.
4.3.
Robustness Issues
Multiple worlds. Our results do not rely on the number of “models of the world” being
maintained by players. Firstly, even when we allow for more than two hypotheses
being maintained, it is still the most extreme one among these hypotheses that will
gain the most credibility following a rare success in another country. To see this, suppose in addition to the tranquility world and the frenzy world, there is also a middle
22
world, M, in which it is less difficult to overthrow a regime than in tranquility world,
but more difficult than in frenzy world. Equilibrium in such a model has the property
that the success probabilities in these three worlds are ordered: p T < p M < p F . Upon
observing a success, Bayes’ rule dictates that the belief in the T-world, π T , must fall
and π F must rise, while π M may increase or decrease. Importantly, since p M < p F ,
the posterior ratio of π F /π M must be higher than its prior ratio. In other words, a
successful attack is evidence in favor of the frenzy world relative to the middle world.
Secondly, our results continue to hold in a more general setting where some of
the possible worlds maintain that the world is even more tranquil than what the true
model implies. We establish this point by considering a general model of model uncertainty in which players maintain a wide range of alternative hypotheses surrounding
the true one. Specifically, suppose there are K worlds, ordered such that p1 < . . . < pK ,
with k∗ being the true model. After observing a success, the posterior belief in the
∗
true hypothesis, π k , may go up or down. However, the posterior belief distribution
π = (π 1 , . . . , π K ) must dominate the prior belief distribution according to the likelihood ratio order, because for any k > l,
πtk+1
πtl +1
=
πtk pk
πtk
.
>
πtl pl
πtl
Further, if θ is between θl∗ and θl∗+1 , then defense is worthwhile for the regime only if
the state is greater than the true survival threshold. Since θ1∗ < . . . < θK∗ , the regime
defends if and only if
l
∑ π k V ≥ d.
k =1
Hence, the payoff from defense falls and the regime is less likely to defend. This in
turn implies that, from the perspective of agent i, the probability of successful attack,
Pk (θ∗ , xi , π ), increases for each k. Moreover the expected payoff from attack is:
K
∑ π k Pk (θ∗ , xi , π ).
k =1
Since P1 < . . . < PK , her expected payoff increases following a successful attack. Thus,
even in this more general framework where players believe that the world can be more
tranquil than it really is, a successful attack in one country always causes agents to become more aggressive and hence raises the probability of success in another country.27
Hypothesis testing. In our model, players are fully Bayesian and they maximize
27 However, we need at least one hypothesis in which the world is more frantic than it truly is in order
to generate a quantitatively significant contagion effect. If k∗ = K, then successes would be rare even if
players fully believe in world K.
23
expected utility taking into account all possible models of the world. Alternatively,
we may assume that players only hold one model at a time when making a decision,
but they also maintain a belief over the model space. They update their belief over
possible models in light of the observed evidence, and keep using the existing model
as long as the belief in this model does not fall below a certain threshold. When the
belief in the existing model falls below the threshold, they pick an alternative model
with the highest belief. Ortoleva (2012) calls this a “hypothesis-testing representation”
and provides an axiomatic foundation for such a decision rule. We argue that the
contagion mechanism in this paper still operates if players adopt a decision rule that
satisfies the hypothesis-testing representation, but the onset and the end of the crisis
period will be even more abrupt than in our model.
To see this, consider a hypothesis-testing representation in which the tranquility
hypothesis is maintained if π > π̂. As long as π > π̂, equilibrium is characterized by
( x ∗ , θ T∗ ) (whose values are equal their counterparts in our model with π = 1) and the
associated p T . But players can also compute what the equilibrium ( x ∗ , θ F∗ ) would be
(which is equal to their counterparts in our model with π = 0) and the associated p F
in the frenzy world. We can assume that players in another country update their belief
about possible worlds before they choose their actions. Proposition 2 still holds. For k T
sufficiently large, p T /p F is small, so that observing a rare success in another country
can push players’ belief in the tranquility world below π̂, and they switch to adopt
the frenzy model as a result. Under this decision rule, the equilibrium probability
of success can only take two values, depending on whether π is above or below the
threshold. The period when it takes the higher value can be considered a crisis period.
Information of the regime. In the benchmark model, we assume that the regime
knows its own strength θ but does not know the true model of the world. As we
have shown, in this case complementarity arises between the actions of agents and
the regime. In a sense, how strong the regime is depends not only the fundamental
strength θ, but also its willingness to fight the attackers. Thanks to this additional layer
of complementarity, the regime can become more fragile simply because agents tend
to believe it is easier to topple.
Alternative assumptions about what the regime knows deliver the same qualitative
results. Suppose, for example, that the regime neither knows θ nor the true model.
Then it chooses to defend if and only if
∗
∗
θF − µ
θT − µ
+ (1 − π ) 1 − Φ
V ≡ δ(θ T∗ , θ F∗ , π ).
d ≤ π 1−Φ
σ
σ
When π declines, δ decreases. In other words, ex ante, it is more likely to observe the
regime to give up when the public belief is revised downwards.
24
Next, suppose the regime knows both θ and the true model of the world. The
regime would defend if and only if θ ≥ θ j∗ , where world j is the true world. Thus, from
the perspective of an agent with private information x, she believes that the probability
p
of successful attack is PT = Φ (θ T∗ − X )/( βσx ) in world T, while PF remains the
same as in the benchmark case. Under this set of assumptions, because the regime
never gives up unless fighting is futile, the equilibrium is the same as in a global game
model where the regime is not an active player. There would be no complementarity
between the actions of the agents and the regime, but the complementarity among the
actions of the agents themselves remains.
Finally, suppose that the regime does not know θ but knows the true model of the
world. Then the regime chooses to defend if and only if (1 − Φ((θ j∗ − µ)/σ ))V ≥ d.
From the perspective of an agent with private information x, an attack would fail in
world j if θ > θ j∗ and the regime chooses to defend. Therefore, for j = T, F, we have
Pj = 1 −
1−Φ
θ j∗ − X
p
βσx
!!
H
1−Φ
θ j∗ − µ
!! !
σ
V
.
Note that Pj is increasing in θ j∗ . Therefore, we still have PT < PF (and, similarly,
p T < p F ). Therefore, a successful attack is evidence against a tranquility world, and
encourages agents in other countries to attack more aggressively. The contagion mechanism works in a way very similar to our benchmark model.
Defense cost. In the benchmark case, the assumption that the defense cost of each
country, d, is independent and drawn from a known distribution H is introduced for
realism. An alternative assumption is that the defense cost may be the same across
countries but agents do not know its exact value and they need to update their beliefs,
based on outcomes in previous countries. We show that this additional mechanism
can strengthen our results quantitively without affecting them qualitatively.
To see this point, suppose the defense cost distribution perceived by the agents
in country t is Ht (d). On condition that there is a success in country t, the density
function of defense cost is updated by Bayes’ rule:

∗ ηΦ θ F −µ ht (d)
σ
h
i
h t +1 ( d | S t = 1 ) =
η (1 − π )Φ θ F∗ −µ + π Φ θT∗ −µ h (d)
t
t
t
σ
σ
if d ≥ πt V,
if d < πt V;
where η is a normalizing constant. Since θ F∗ > θ T∗ , observing St = 1 induces a firstorder stochastic increase in the distribution Ht+1 (·|St = 1) relative to Ht (·). Intuitively,
upon a success observed in country t, agents believe that the defense cost is more likely
to be higher than πt V and less likely to be smaller than that. Such updating makes
25
agents in country t + 1 even more aggressive, because they believe that the probability
of the regime choosing defense, Ht+1 (πt+1 V ), is even smaller.
5.
Concluding Remarks
It is not difficult to understand why people can drastically change their belief and
hence their behavior following a rare event. A case in point is the plummeting of air
travel shortly after the September 11 attack, which took years to recover to the preattack level.28 Arguably, that was the safest time to travel because airport security
was extremely tight. Nevertheless, our theory suggests that it was a rational response
for people to avoid air travel: the totally unexpected terrorist attack had shaken their
belief that the air travel system is secure. They tend to believe the system is risk-prone
and need to accumulate further evidence to restore their confidence.
In the case of air travel, an increase in perception of risks was associated with increased efforts to counteract those risks. So beliefs and actions were not reinforcing
each other. That is probably why we did not observe a wave of terrorist attacks of
the same type. But in coordination games such as currency attacks, mass revolutions,
and bank runs, an increase in the perception of fragility of the regime can lead to increased efforts to topple it. A successful attack elsewhere does raise such a perception
dramatically for subsequent similar regimes. In sum, we show that the rarity of a
single crisis, coupled with the complementarity between actions and beliefs, can be a
powerful contagion mechanism and produce a wave of crises.
In a sense this paper attempts to revive the role of belief fragility in our thinking
about currency crises or revolutions, which was central for the “second generation”
of crisis models. In a standard global game framework, the fundamental strength
of the regime is the key to the equilibrium outcome. In our model, the belief over
possible models of world is as critical as the fundamental strength. A sudden shift
in this belief can also arise, but it is provoked by rational belief updating over the
model space, rather than a switch or jump between multiple equilibria. Further, since
our model retains the advantage of having a unique equilibrium, we can bypass the
issue of equilibrium selection and therefore offer an explicit link for the evolution of
equilibrium.
28 See
“How 9/11 Changed America: in Statistics,” BBC News, 1 September, 2006.
26
Appendix
Proof of Proposition 1. First, we assume a monotone equilibrium and show that it is
unique when the condition holds that D 0j (θ )φ(0)σx < σ2 . Given x ∗ , let θ j∗ ( x ∗ ) satisfy
the critical mass conditions (2) and (3). Differentiate these equations with respect to
x ∗ , we obtain
dθ j∗
dx ∗
=
1
σx φ
x ∗ −θ j∗
σx
D 0j (θ j∗ ) + σ1x φ
x ∗ −θ j∗
σx
> 0.
Since φ(t) attains at maximum at t = 0, dθ j∗ /dx ∗ is bounded above by φ(0)/( D 0j (θ j∗ )σx +
φ(0)), which is less than β = σ2 /(σx2 + σ2 ).
Let g( x ) = πPT (θ T∗ ( x ), θ F∗ ( x ), x ) + (1 − π ) PF (θ F∗ ( x ), x ) − c. This function is strictly
decreasing because dθ j∗ /dx ∗ < β. Moreover, limx→∞ g( x ) = −c and limx→−∞ g( x ) =
1 − c. Therefore, for any c ∈ (0, 1), there exists a unique x ∗ that satisfies g( x ∗ ) = 0.
According to the critical mass conditions (2) and (3), for any x ∗ , there exists a unique
pair (θ T∗ , θ F∗ ).
Second, we show that non-monotone equilibria do not exist. Given a mass of at1
tackers A, the attack succeeds in world j if θ ≤ D −
j ( A ). The regime’s best response to
A is to choose y( A, θ ) = 1 if (a) θ > DF−1 ( A), or (b) θ > DT−1 ( A) and d < πV; and to
choose y( A, θ ) = 0 otherwise. Given A and the regime’s strategy y(·), the gain from
attack for agent i is
G ( A, xi ) = π Pr[θ ≤ DT−1 ( A) or y( A, θ ) = 0 | xi ]) + (1 − π ) Pr[θ ≤ DF−1 ( A) | xi ] − c.
1
Since D −
j (·) is increasing and y (·, θ ) is non-increasing, G ( A, xi ) increases in A. Since
y( A, θ ) is non-decreasing in θ and the distribution of θ is stochastically increasing in
xi , G ( A, xi ) decreases in xi . Let A(τ ) represent the aggregate mass of attackers when
agents adopt strategy τ, and consider the monotone strategy τκ , with τκ ( xi ) = 1 if
xi < κ and τκ ( xi ) = 0 if xi > κ. Then G ( A(τκ ), κ ) = g(κ ), and we have already shown
that there is a unique x ∗ such that g( x ∗ ) = 0. Therefore, the monotone strategy τx∗ is
an equilibrium strategy.
Now, consider any equilibrium strategy τ 0 of the game. Let x be the smallest xi
such that G ( A(τ 0 ), xi ) ≤ 0 and let x be the largest xi such that G ( A(τ 0 ), xi ) ≥ 0. We
have x ≤ x. Since G ( A(τ 0 ), xi ) > 0 for any xi < x, we have τ 0 ( xi ) = 1 for xi < x.
Thus, A(τ 0 ) ≥ A(τx ), which in turn implies g( x ) = G ( A(τx ), x ) ≤ G ( A(τ 0 ), x ) ≤ 0.
But since g(·) is decreasing, we must have x ≥ x ∗ . A parallel argument establishes
that x ≤ x ∗ . Together, they imply x = x = x ∗ . Thus, τ 0 = τx∗ . In other words, any
equilibria except the monotone equilibrium do not exist.
27
Use the implicit theorem and differentiate equation g(·) with respect to π to obtain:
πVh(πV )
dx ∗
= ( PF − PT ) 1 +
.
g (x )
dπ
H (πV )
0
∗
Since g0 (·) < 0 and PF > PT , dx ∗ /dπ < 0. Moreover, for j = T, F, dθ j∗ /dπ =
(dθ j∗ /dx ∗ )(dx ∗ /dπ ) < 0, which in turn implies dp j /dπ < 0.
Proof of Proposition 2. Write g( x; k T ) = πPT (θ T∗ ( x; k T ), θ F∗ ( x ), x ) + (1 − π ) PF (θ F∗ ( x ), x ) −
c, where θ j∗ (·) solves the respective critical mass condition in world j. Implicit differentiation of the indifference condition g( x ∗ ; k T ) = 0 gives
dx ∗
∂PT ∂θ T∗
= −π ∗
.
g (x )
dk T
∂θ T ∂k T
0
∗
Thus dx ∗ /dk T < 0, because g0 < 0, ∂PT /∂θ T∗ < 0, and ∂θ T∗ /∂k T < 0. This in turn
implies:
dθ T∗
dθ T∗ dx ∗
∂θ T∗
=
+
< 0,
dk T
dx dk T
∂k T
dθ F∗
dθ ∗ dx ∗
= F
< 0.
dk T
dx dk T
Now, combine the critical mass conditions in the two worlds to obtain:
Φ
x ∗ − θ T∗
σx
−Φ
x ∗ − θ F∗
σx
= DT (θ T∗ ; k T ) − DF (θ F∗ ; k F ).
Since the left-hand-side of the above is bounded, we must have θ F∗ − θ T∗ going to infinity as k T goes to infinity. Let ĝ( x ) = (1 − π ) PF (θ F∗ ( x ), x ) − c and let x solves ĝ( x ) = 0.
Since ĝ( x ) < g( x ) for any x, we have x ∗ > x for any k T , and thus θ F∗ > θ F∗ ( x ) for any
k T . Thus, as k T goes to infinity, we must have θ T∗ going to negative infinity, while θ F∗
and x ∗ remaining bounded.
Because p T is increasing in θ T∗ and θ F∗ , which are decreasing in k T , we have dp T /dk T <
0. This proves part (a). Further,
dθ T∗
dθ ∗
− F =
dk T dk T
dθ T∗
dθ ∗
− F
dx
dx
∂θ ∗
dx ∗
+ T.
dk T
∂k T
Because x ∗ is decreasing but remains bounded as k T increases, we have dx ∗ /dk T going
to 0 for k T sufficiently large. On the other hand, ∂θ T∗ /∂k T is negative and bounded
away from 0. Therefore, dθ T∗ /dk T < dθ F∗ /dk T for k T large. The derivative of p T /p F
28
with respect to k T has the same sign as:
θ T∗ −µ
σ
θ T∗ −µ
φ
Φ
σ
θ F∗ −µ
σ
φ
dθ T∗
dθ ∗
− θ ∗ −µ F .
dk T
dk T
Φ Fσ
The function φ(·)/Φ(·) is decreasing, and dθ T∗ /dk T < dθ F∗ /dk T < 0. Therefore p T /p F
decreases in k T for k T sufficiently large. This proves part (b). As k T goes to infinity,
p T /p F approaches 1 − H (πV ). The limit value of πt+1 follows from the Bayes’ formula.
29
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